Research On The Characteristics Of A Class Of High-Dimensional Non-Degenerate Discrete Chaotic Maps

Encryption of information using chaos theory is a proven way in the field of communication security.However,with the development of chaos identification technology,some low-dimensional chaotic systems with simple dynamic behavior are vulnerable to attacks such as inverse iteration,spectrum analysis,phase space reconstruction and so on.Therefore,it is an important and challenging task to study chaotic systems with richer and more complex dynamical behaviors.Nonlinear systems can be classified into continuous systems and discrete systems,where discrete systems are faster to iterate and easier to implement on digital hardware platforms.In addition,compared to low dimensional maps or degenerate maps,high-dimensional non-degenerate maps of two-dimensional and above have better chaotic performance.Therefore,this dissertation discusses the design,analysis and implementation of a class of high-dimensional non-degenerate chaotic maps with initial-boosting behavior around discrete nonlinear circuits and systems,and conducts an in-depth study of some of the properties and phenomena.The following three main areas of work are accomplished:Firstly,a new discrete memristor model is designed based on periodic triangular wave function,and a two-dimensional and a three-dimensional discrete memristive map is constructed respectively,and the dynamic characteristics are analyzed in detail.The two-dimensional memristive map has a simple structure,can produce hyperchaotic attractor,and is a non-degenerate map.Three-dimensional memristive maps have hidden attractors and the number of positive Lyapunov exponents can reach the maximum ideal state.And they all have an initial-boosting behavior that depends on the initial value of the memristor,and can generate an infinite number of homogeneous coexisting attractors,and their physical realizability is proved by using DSP hardware platform.Secondly,aiming at the problem that the chaotic region of the discrete memristive chaotic map is too small,a new three-dimensional discrete chaotic map with infinitely wide parameter range is proposed by using the same periodic trigonometric wave function as above.It has been mathematically proven that the map can maintain a non-degenerate hyperchaotic state within an infinite wide parameter range under certain conditions.Similarly,the three-dimensional map also has an initial-boosting behavior that depends on the initial value,which can generate an infinite number of homogenous coexistence attractors and implement them physically.Finally,in order to expand the chaotic map with limited dimensions,a new construction method of N-dimensional non-degenerate discrete chaotic map is proposed by using the same periodic trigonometric wave function.This method can construct two-dimensional and above discrete chaotic maps with non-degeneracy properties.The concept of seed function is introduced to make the maps form have diversity.At the same time,it is proved mathematically that the chaotic map constructed by this method is non-degenerate.As an example,three sub-maps are constructed respectively,and the results showed that regardless of the form of the seed function,the sub-maps are non-degenerate.Similarly,the sub-map also has an initial-boosting behavior that depends on the initial value,but unlike the above,the coexistence attractor can be heterogeneous,and the sub-map is physically implemented.The design and analysis of chaotic maps with complex dynamic behaviors is the first and crucial step in the practical application of chaos theory.In this dissertation,a class of discrete chaotic mapping with no degeneracies is constructed step by step,its dynamic behavior is analyzed,deeply analyzes several phenomena and characteristics,and uses a digital microcontroller platform for physical implementation,which will provide an effective theoretical premise for further applications of discrete chaotic maps in the field of secure communications.